In this talk we will discuss epsilon-periodic composite media and their limiting properties (as epsilon goes to zero) from the point of view of waves propagation. Such problems appear in different areas of materials science in different setups. We will concentrate on the most interesting case, the case of so-called high-contrast media. Such models exhibit unusual behavior (in particular, metamaterial one) at certain wavelengths and are characterized by a special relationship between the contrast and the period length. I will present an operator-theoretic approach to the derivation of an asymptotically equivalent (admitting sharp norm-resolvent convergence estimates) model, which allows us to determine the bands where the group velocity becomes negative. We will demonstrate metamaterial behavior in an example of a thin periodic network shrinking to a quantum graph.

The talk will be centred around the recent paper Cherednichenko, K., Kiselev, A.V., Velčić, I. et al. Effective Behaviour of Critical-Contrast PDEs: Micro-Resonances, Frequency Conversion, and Time Dispersive Properties. II. Commun. Math. Phys. 406, 72 (2025). doi. I will discuss our recent progress in this area, paying special attention to the following two aspects: (i) how the critical contrast setup almost automatically leads to time-dispersive models, and (ii) how the latter can be achieved without the former. Time permitting, I will strive to at least mention in the passing the non-periodic case, in particular having random displacement Anderson-type models in mind. The first half of the talk will be devoted to the introduction of the necessary concepts of operator and spectral theory, such as generalised resolvents, Dirichlet-to-Neumann maps, and conservative dilations, and will be thus accesible to an audience well-versed in modern mathematical physics.

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